Tài liệu Đề tài " Weak mixing for interval exchange transformations and translation flows " ppt

Annals of Mathematics Weak mixing for interval exchange transformations and translation flows By Artur Avila and Giovanni Forni*... Weak mixing for interval exchangetransformations an

Annals of Mathematics

Weak mixing for interval

exchange transformations and

translation flows

By Artur Avila and Giovanni Forni*

Weak mixing for interval exchange

transformations and translation flows

By Artur Avila and Giovanni Forni*

Abstract

We prove that a typical interval exchange transformation is either weaklymixing or it is an irrational rotation We also conclude that a typical trans-

lation flow on a typical translation surface of genus g ≥ 2 (with prescribed

singularity types) is weakly mixing

1 Introduction

Let d ≥ 2 be a natural number and let π be an irreducible permutation

of {1, , d}; that is, π{1, , k} = {1, , k}, 1 ≤ k < d Given λ ∈Rd

+, we

define an interval exchange transformation (i.e.t.) f := f (λ, π) in the usual

way [CFS], [Ke]: we consider the interval

We are interested in the ergodic properties of i.e.t.’s Obviously, they preserve

the Lebesgue measure on I Katok proved that i.e.t.’s and suspension flows over

*A Avila would like to thank Jean-Christophe Yoccoz for several very productive discussions and Jean-Paul Thouvenot for proposing the problem and for his continuous encouragement A Avila

is a Clay Research Fellow G Forni would like to thank Yakov Sinai and Jean-Paul Thouvenot who suggested that the results of [F1], [F2] could be brought to bear on the question of weak mixing for i.e.t.’s G Forni gratefully acknowledges support from the National Science Foundation grant DMS-0244463.

i.e.t.’s with roof function of bounded variation are never mixing [Ka], [CFS].Then the fundamental work of Masur [M] and Veech [V2] established thatalmost every i.e.t is uniquely ergodic (this means that, for every irreducible

+, f (λ, π) is uniquely ergodic).

The question of whether the typical i.e.t is weakly mixing is more delicate

except if π is a rotation of {1, , d}, that is, if π satisfies the following

condi-tions: π(i + 1) ≡ π(i) + 1 mod d, for all i ∈ {1, , d} In that case f(λ, π) is

conjugate to a rotation of the circle, hence it is not weakly mixing, for every

λ ∈Rd

+ After the work of Katok and Stepin [KS] (who proved weak mixing foralmost all i.e.t.’s on 3 intervals), Veech [V4] established almost sure weak mix-ing for infinitely many irreducible permutations and asked whether the sameproperty is true for any irreducible permutation which is not a rotation Inthis paper, we give an affirmative answer to this question

Theorem A Let π be an irreducible permutation of {1, , d} which is

+, f (λ, π) is weakly mixing.

We should remark that topological weak mixing was established earlier

(for almost every i.e.t which is not a rotation) by Nogueira-Rudolph [NR]

We recall that a measure-preserving transformation f of a probability space (X, m) is said to be weakly mixing if for every pair of measurable sets A,

It follows immediately from the definitions that every mixing transformation

is weakly mixing and every weakly mixing transformation is ergodic A

clas-sical theorem states that any invertible measure-preserving transformation f

is weakly mixing if and only if it has continuous spectrum; that is, the only eigenvalue of f is 1 and the only eigenfunctions are constants [CFS], [P] Thus

it is possible to prove weak mixing by ruling out the existence of non-constant

measurable eigenfunctions This is in fact the standard approach which is also

followed in this paper Topological weak mixing is proved by ruling out the

ex-istence of non-constant continuous eigenfunctions Analogous definitions and

statements hold for flows

1.1 Translation flows Let M be a compact orientable translation surface

of genus g ≥ 1, that is, a surface with a finite or empty set Σ of conical

singularities endowed with an atlas such that coordinate changes are given bytranslations inR 2 [GJ1], [GJ2] Equivalently, M is a compact surface endowed

with a flat metric, with at most finitely many conical singularities and trivialholonomy For a general flat surface the cone angles at the singularities are

satisfying 

for all 1≤ i ≤ r, and there exists a parallel section of the unit tangent bundle

A third, equivalent, point of view is to consider pairs (M, ω) of a compact Riemann surface M and a (non-zero) abelian differential ω A flat metric on

v of unit length is determined by the condition ω(v) = 1 The specification of

the parameters κ = (κ1, , κr) ∈Zr

+ with 

dimensional stratum H(κ) of the moduli space of translation surfaces which is

endowed with a natural complex structure and a Lebesgue measure class [V5],[Ko]

A translation flow F on a translation surface M is the flow generated by

a parallel vector field of unit length on M \ Σ The space of all translation

flows on a given translation surface is naturally identified with the unit tangent

space at any regular point; hence it is parametrized by the circle S1 For all

θ ∈ S1, the translation flow F θ , generated by the vector field v θ such that

e −iθ ω(v θ) = 1, coincides with the restriction of the geodesic flow of the flatmetric|ω| to an invariant surface M θ ⊂ T1M (which is the graph of the vector

field v θ in the unit tangent bundle over M \ Σ).

We are interested in typical translation flows (with respect to the Haar measure on S1) on typical translation surfaces (with respect to the Lebesgue

measure class on a given stratum) In genus 1 there are no singularities andtranslation flows are linear flows on T 2: they are typically uniquely ergodic

but never weakly mixing In genus g ≥ 2, the unique ergodicity for a typical

translation flow on the typical translation surface was proved by Masur [M]and Veech [V2] This result was later strenghtened by Kerckhoff, Masur andSmillie [KMS] to include arbitrary translation surfaces

As in the case of interval exchange transformations, the question of weakmixing of translation flows is more delicate than unique ergodicity, but it is

widely expected that weak mixing holds typically in genus g ≥ 2 We will show

that it is indeed the case:

Theorem B Let H(κ) be any stratum of the moduli space of translation

Translation flows and i.e.t.’s are intimately related: the former can beviewed as suspension flows (of a particular type) over the latter However, sincethe weak mixing property, unlike ergodicity, is not invariant under suspensionsand time changes, the problems of weak mixing for translation flows and i.e.t.’sare independent of one another We point out that (differently from the case ofi.e.t.’s, where weak mixing had been proved for infinitely many combinatorics),

there has been little progress on weak mixing for typical translation flows

(in the measure-theoretic sense), except for topological weak mixing, proved

in [L] Gutkin and Katok [GK] proved weak mixing for a G δ-dense set oftranslation flows on translation surfaces related to a class of rational polygonalbilliards We should point out that our results tell us nothing new aboutthe dynamics of rational polygonal billiards (for the well-known reason thatrational polygonal billiards yield zero measure subsets of the moduli space ofall translation surfaces)

1.2 Parameter exclusion To prove our results, we will perform a

param-eter exclusion to get rid of undesirable dynamics With this in mind, instead

of working in the direction of understanding the dynamics on the phase space(regularity of eigenfunctions1, etc.), we will focus on analysis of the parameterspace

We analyze the parameter space of suspension flows over i.e.t.’s via arenormalization operator (i.e.t.’s correspond to the case of constant roof func-tion) The renormalization operator acts non-linearly on i.e.t.’s and linearly onroof functions, so it has the structure of a cocycle (the Zorich cocycle) over therenormalization operator on the space of i.e.t.’s (the Rauzy-Zorich induction).One can work out a criterion for weak mixing (originally due to Veech [V4])

in terms of the dynamics of the renormalization operator

An important ingredient in our analysis is the result of [F2] on the uniform hyperbolicity of the Kontsevich-Zorich cocycle over the Teichm¨ullerflow This result is equivalent to the non-uniform hyperbolicity of the Zorich co-cycle [Z3] Actually we only need a weaker result, namely that the Kontsevich-Zorich cocycle, or equivalently the Zorich cocycle, has two positive Lyapunovexponents in the case of surfaces of genus at least 2

non-In the case of translation flows, a “linear” parameter exclusion (on theroof function parameters) shows that “bad” roof functions form a small set(basically, each positive Lyapunov exponent of the Zorich cocycle gives oneobstruction for the eigenvalue equation, which has only one free parameter).This argument is explained in Appendix A

The situation for i.e.t.’s is much more complicated, since we have no dom to change the roof function We need to carry out a “non-linear” exclusionprocess, based on a statistical argument This argument proves weak mixing

free-at once for typical i.e.t.’s and typical translfree-ation flows While for the linearexclusion it is enough to use the ergodicity of the renormalization operator

on the space of i.e.t.’s, the statistical argument for the non-linear exclusionheavily uses its mixing properties

1 In this respect, we should remark that Yoccoz has pointed out to us the existence of “strange” eigenfunctions for certain values of the parameter.

1.3 Outline We start this paper with basic background on cocycles

over expanding maps We then prove our key technical result, an abstractparameter exclusion scheme for “sufficiently random integral cocycles”

We then review known results on the renormalization dynamics for i.e.t.’sand show how the problem of weak mixing reduces to the abstract parameterexclusion theorem The same argument also covers the case of translationflows

In the appendix we present the linear exclusion argument, which is muchsimpler than the non-linear exclusion argument but is enough to deal withtranslation flows and yields an estimate on the Hausdorff dimension of the set

of translation flows which are not weakly mixing

2 Background

2.1 Strongly expanding maps Let (∆, µ) be a probability space We say that a measurable transformation T : ∆ → ∆, which preserves the measure

class of the measure µ, is weakly expanding if there exists a partition (modulo

0){∆ (l) , l ∈Z} of ∆ into sets of positive µ-measure, such that, for all l ∈ Z,

T maps ∆ (l) onto ∆, T (l) := T |∆ (l) is invertible and T ∗ (l) µ is equivalent to µ.

Let Ω be the set of all finite words with integer entries The length (number

of entries) of an element l ∈ Ω will be denoted by |l| For any l ∈ Ω, l =

projectiviza-tion of Rp+ We will call it the standard simplex A projective contraction is

a projective transformation taking the standard simplex into itself Thus a

projective contraction is the projectivization of some matrix B ∈ GL(p,R)with non-negative entries The image of the standard simplex by a projective

contraction is called a simplex We need the following simple but crucial fact.

lemma 2.1 Let ∆ be a simplex compactly contained in Pp −1

invertible and its inverse is the restriction of a projective contraction Then T preserves a probability measure µ which is absolutely continuous with respect to the Lebesgue measure on ∆ and has a density which is continuous and positive

on ∆ Moreover, T is strongly expanding with respect to µ.

Proof Let d([x], [y]) be the projective distance between [x] and [y]:

densities have logarithms which are p-Lipschitz with respect to the projective

distance Since ∆ has finite projective diameter, it suffices to show that there

exists µ ∈ N invariant under T and such that µl ∈ N for all l ∈ Ω Notice

Since (Tl)−1 is the projectivization of some matrix Bl= (blij ) in GL(p,R)with non-negative entries, we have

Let µ be any limit point of {ν n } in the weak* topology Then µ is invariant

under T , belongs to N and, for any l ∈ Ω, µl is a limit of

2.3 Cocycles A cocycle is a pair (T, A), where T : ∆ → ∆ and A : ∆ →

GL(p,R), viewed as a linear skew-product (x, w) → (T (x), A(x)·w) on ∆×Rp

Notice that (T, A) n = (T n , A n), where

If (T, A) is a measurable cocycle, the Oseledets Theorem [O], [KB] implies

that lim n (x) 1/n exists for almost every x ∈ ∆ and for every w ∈ Rp,

and that there are p Lyapunov exponents θ1 ≥ · · · ≥ θ p characterized by

(2.14) #{i, θ i = θ } = dim{w ∈Rp , lim n (x) 1/n ≤ e θ }

, lim n (x) 1/n < e θ }

Thus dim E cs (x) = # {i, θ i ≤ 0}.2 Moreover, if λ < min {θ i , θ i > 0} then for

almost every x ∈ ∆, for every subspace G0 ⊂ Rp transverse to E cs (x), there exists C(x, G0) > 0 such that

we will call (T, A) a uniform cocycle.

2It is also possible to show that dim E s (x) = # {i, θ < 0 }.

lemma 2.2 Let (T, A) be a uniform cocycle and let

Notice that T ∗ B κ ⊂ B κ Let

(2.20) ω N (κ) = sup

ν ∈B κ

1

The result follows from (2.21), (2.22) and (2.23)

We say that a cocycle (T, A) is locally constant if T : ∆ → ∆ is strongly

expanding and A |∆ (l) is a constant A (l) , for all l ∈Z In this case, for all l∈ Ω,

l = (l1, , ln), we let

(2.24) Al:= A (l n)· · · A (l1)

.

We say that a cocycle (T, A) is integral if A(x) ∈ GL(p,Z), for almost all

3 Exclusion of the weak-stable space

Let (T, A) be a cocycle We define the weak-stable space at x ∈ ∆ by

(3.1) W s (x) = {w ∈Rp , n (x) Rp /Zp → 0} ,

where Rp /Zp denotes the euclidean distance from the lattice Zp ⊂Rp Now,

it is immediate to see that, for almost all x ∈ ∆, the space W s (x) is a union of

translates of E s (x) If the cocycle is integral, W s (x) has a natural tion as the stable space at (x, 0) of the zero section in ∆ ×Rp /Zp If the cocycle

interpreta-is bounded, that interpreta-is, if the function A : ∆ → GL(p,R) is essentially bounded,

then it is easy to see that W s (x) = ∪ c ∈Zp E s (x) + c In general W s (x) may be the union of uncountably many translates of E s (x).

Let Θ⊂Pp −1 be a compact set We say that Θ is adapted to the cocycle

(T, A) if A (l) · Θ ⊂ Θ for all l ∈Zand if, for almost every x ∈ ∆,

(3.2)

whenever w ∈Rp \ {0} projectivizes to an element of Θ.

not passing through 0

The main result in this section is the following

theorem 3.1 Let (T, A) be a locally constant integral uniform cocycle

J ∩ E cs (x) = ∅ for almost every x ∈ ∆ Then if L is a line contained in Rp

 1+ε

transparent already under the condition  ε

which we will apply Theorem 3.1 in this paper, namely, uniformly hyperbolicinducings of the Zorich cocycle, it is well known that 

was recently shown in [AGY] that one can choose the cocycles so as to obtain

every x Thus, for every δ > 0 and J ∈ J , there exists C δ (J ) > 0 such that

C δ (J ) } < δ for every J  in a neighborhood of J By compactness, there exists

C δ > 0 such that µ{x, C(x, J) ≤ C δ

hence for every J ∈ J The result now follows since 2ε0 < λ.

For any δ < 1/10, let W s

δ,n (x) be the set of all w ∈ B δ(0) such that

k (x) Rp /Zp < δ for all k ≤ n, and let W s

compo-nents of the set Al(J ∩ B δ(0))∩ B δ(Zp \ {0}) and let φ δ(l) := supJ ∈J φ δ (l, J ).

For any (fixed) l∈ Ω the function δ → φ δ(l) is non-decreasing and there exists

δl> 0 such that for δ < δl we have φ δ(l) = 0 We also have

Given J with l,1 , , J l,φ δ (l,J) be all the lines of the

form Al · J − c where Al(J ∩ B δ(0)) ∩ B δ (c) = ∅ with c ∈ Zp \ {0} Let

For N ∈N\ {0}, let Ω N be the set of all words of length N , and ΩN be

the set of all words of length a multiple of N

For any 0 < η < 1/10, select a finite set Z ⊂ Ω N such that µ( ∪l∈Z∆l) >

1− η Since the cocycle is locally constant and uniform, there exists 0 < η0 <

1/10 such that, for all η < η0,

l∈Ω N \Z

ln l 0µ(∆l) < 1

10 .

claim 3.5 There exists N0 ∈ N\ {0} such that, if N > N0 , then for

that the following holds For every N > N0(κ) and every J ∈ J there exists

Z  := Z  (N, J ) ⊂ Z such that, for all l ∈ Z ,

Hence the claim is proved since N0 ≥ max{N0 (κ), 4ε −10 }.

claim 3.6 Let N > N0 There exists ρ0 (Z) > 0 such that, for every

0 < ρ < ρ0(Z), every J ∈ J and every Y ⊂ ∆ with µ(Y ) > 0,

At this point we fix 0 < η < η0, N > N0, Z ⊂ Ω N , and 0 < ρ < ρ0(Z) so that (3.13) and (3.20) hold and let δ < 1/10 be so small that

µ (Γ m δ (J )) → 0 for every J ∈ J Let Ω be, as above, the set of all finite words

with integer entries Let ΩN and ΩN be, as above, the subset of all words of

length N and the subset of all words of length a multiple of N , respectively Let ψ : Ω N →Zbe such that ψ(l) = 0 if l ∈ Z and ψ(l) = ψ(l ) whenever l= l 

and l / ∈ Z We let Ψ : Ω N → Ω be given by Ψ(l(1) l (m) ) = ψ(l(1)) ψ(l (m)),

where the l(i) are in ΩN We let ∆d =∪l∈Ψ −1(d)∆l

For d∈ Ω, let C(d) ≥ 0 be the smallest real number such that

Proof Let d = (d1, , d m+1), d = (d2, , dm+1) There are two

possi-bilities: (1) If d1 = 0, we have by (3.20) and (3.24)